
where \(J_{[\,]}(1,\mathcal {F},L_2(\mathbb {P}_n) ) = \int _{0}^1 \sqrt{ 1+ \log N_{[\, ]}(\varepsilon \Vert F_q\Vert ,\mathcal {F},L_2(\mathbb {P}_n)) }d\varepsilon \)  is the bracketing integral of the functions class \(\mathcal {F} = \lbrace  \theta \rightarrow \partial _\theta q(z_i,\theta ) \rbrace \)  and \(N_{[\, ]}(\varepsilon \Vert F_q\Vert ,\mathcal {F},L_2(\mathbb {P}_n))\)  its bracketing number. By Theorem 9.23 in [1]}, we have \(N_{[\, ]}(\varepsilon \Vert F_q\Vert ,\mathcal {F},L_2(\mathbb {P}_n)) \le N(\varepsilon /2,\Theta ,\Vert \cdot \Vert ) \le (6/\varepsilon )^{d_\theta } \text{vol}(\Theta )/\text{vol}(B)\)  where \(\text{vol}(\Theta ),\text{vol}(B)\)  are the volumes of \(\Theta \)  and the unit sphere \(B\)  in \(\mathbb {R}^{d_\theta }\) , respectively. As a result, we have:
\(&\mathbb {E}^\star \left( \sup _{\theta \in \Theta } \sqrt{m}\Vert G_m^{(b)}(\theta ) - G_n(\theta )\Vert  \right) \lesssim \sqrt{d_{\theta }} \left[ \frac{1}{n} \sum _{i=1}^n |F_q(z_i)|^2 \right]^{1/2},\) 
